Bellman Systems with Mean Field Dependent Dynamics

Citation:

Alain BENSOUSSAN,Miroslav BUL'{I}v{C}EK,Jens FREHSE.Bellman Systems with Mean Field Dependent Dynamics[J].Chinese Annals of Mathematics B,2018,39(3):461~486
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Authors:

Alain BENSOUSSAN; Miroslav BUL'{I}v{C}EK;Jens FREHSE

Foundation:

This work was supported by the National Science Foundation (Nos.DMS 1303775, DMS 1612880), the Hong Kong SAR Research Grant Council (Nos.GRF 500113, 11303316), Hausdorff Center for Mathematics at University of Bonn and the Czech Science Foundation (No.16-03230S).
Abstract: The authors deal with nonlinear elliptic and parabolic systems that are the Bellman like systems associated to stochastic differential games with mean field dependent dynamics. The key novelty of the paper is that they allow heavily mean field dependent dynamics. This in particular leads to a system of PDE's with critical growth, for which it is rare to have an existence and/or regularity result. In the paper, they introduce a structural assumptions that cover many cases in stochastic differential games with mean field dependent dynamics for which they are able to establish the existence of a weak solution. In addition, the authors present here a completely new method for obtaining the maximum/minimum principles for systems with critical growths, which is a starting point for further existence and also qualitative analysis.

Keywords:

Stochastic games, Bellman equation, Mean field equation, Nonlinearelliptic equations, Weak solution, Maximum principle

Classification:

35J60, 35K55, 35J55, 35B65
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